The rider from Sark along with teammates Charlotte Fry and Charlotte Dujardin finished third behind Olympic title holders Germany and runners-up the United States.
It is Britain's third successive team medal in the sport, completing the full set after gold at London 2012 and silver at Rio 2016.
Hester, the oldest member of Team GB at 54 and contesting his sixth Games, gave the team an excellent start as he scored 2577.5 with En Vogue.
Twenty-five-year-old Fry, whose late mother Laura competed for Great Britain at the 1992 Barcelona Olympics, registered 2528.5 aboard Everdale.
Dujardin, riding major championship debutant Gio, anchored the British team’s performance, posting a score of 2617 points for third place, just 24 points overall behind the United States.
It was an impressive achievement by the British trio, with Fry making her Games bow and all three team horses having never previously competed at an Olympics.
'That was cool,' said Hester.
'I had a great balance with En Vogue, and I feel he came here as a boy and he went in there like a man.
'It still feels like he is one of the best horses you could ever sit on. He is amazing.
'We’ve always known he is so talented, and I have just been waiting for the right ride so I could just prove what we know we think he is.'